Number Thirty (30)

Powerful ♦ Pervasive ♦ Profound

Welcome to the most comprehensive review of the number 30 ever created. The number 30 possesses remarkable attributes,
including–and perhaps most profoundly–its role (along with its prime factors 2, 3 and 5) as a primary organizing principle
in the distribution of prime numbers. Before getting to that, here's a list of mathematical properties and other interesting facts
relating to this integer:

30 is the largest number such that all smaller numbers prime relative to it are actually 1 or prime; those numbers
being: 1, 7, 11, 13, 17, 19, 23 & 29. 30 is thus the largest member of the 10-element sequence of
very round numbers (the other 9 elements in this sequence being 1, 2, 3, 4, 6, 8, 12, 18 & 24)
a reduced residue system consisting of only primes and 1. In 30's case the reduced residue is expressed as {30,{1, 7, 11, 13, 17, 19, 23, 29}}.

30 is a semiperfect number
(aka pseudoperfect number) in that a subset of its divisors (5, 10 and 15) = 30.

30 is the total unit length of the straight line segments used to construct a
star polygon
notated as {p/q} where p=10 and q=3 (q is termed the density of the star polygon; in this case the straight lines connect every 3rd point
of 10 equally spaced points lying on a circle's circumference.); thus {10/3}. The
Schläfli symbol for this polygon is {30}. Illustration, below:

30 is the Coxeter Group number h,
dual Coxeter number and the highest degree of fundamental invariance of the Lie Group
E8. You'll note, looking
at the graphical representation of E8 (see image at top of this page) that the perimeters of every one of its
multiple concentric circles possesses 30 points. E8 has 2-, 3- and 5-torsion, and
its exponents are the co-primes up to 30, i.e., 1, 7, 11, 13, 17, 19, 23, and 29.

The graphic below superimposes an image of E8 with a star polygon and the 8 radii of a modulo 30 factorization wheel:

The Number 30 in Lie Group E8: This graphic superimposes images of the star
polygon, modulo 30 wheel factorization radii and "E8 graph of the Gosset 421 polytope as a 2-dimensional skew orthogonal projection
inside Petrie polygon ... an emulation of the hand drawn original by Peter McMullen" licensed by Creative Commons;
license terms here.

The number 30, when plugged into
Euler's totient function, phi(n): phi(30)= 8, with the 8 integers smaller than and having no factors in common with 30 being:
1, 7, 11, 13, 17, 19, 23 & 29. Thirty is the largest integer with this property.

Modulo 30 of all prime numbers (with the exception of 2, 3 and 5) must be 1, 7, 11, 13, 17, 19, 23 or 29.

Below is a magic square where all horizontal, vertical and corner-to-corner
diagonal sums total 30 (created by Gary Croft on 17 March, 2012). Explore its beautiful symmetries involving 10's:

Here's another magic square with rows, columns and principal diagonals totalling 30; in this case the first three prime numbers {2,3,5} are
configured into 3 square matrices (each containing three 2's, 3's and 5's) which in turn are multiplied times three to make magic (created by Gary Croft
on 25 April, 2012):

When arrayed in eight columns, the set of all natural numbers not divisible by 2, 3 and 5 (which by definition
consists of 1 and all prime numbers >5 and their multiples) possesses perfect symmetry involving the number 30, as illustrated below. [Also note that
the sum of the digital root sums (1 + 7 + 2 + 4 + 8 +1 + 5 + 2) of the first 8 elements in this set (1, 7, 11, 13, 17, 19, 23, 29) = 30.]:

When twin primes and twin prime candidates
≥ (11, 13) are arrayed in this divergent (aka harmonic) sequence: 11{+2+4+2+10+2+10} {repeat ... n}, the
intervals between them (2+4+2+10+2+10) total 30, as shown in the matrix below. (Click this link for a deep exploration: twin primes.)

= Prime

= Interval

= Composite

prime

+2

prime

= Prime Pair

30 is interesting when contextualized within the set of natural numbers 1-100 (Hint: Start with the bottom row, then work up.):

30 is the total number of major and minor keys in Western tonal music, including enharmonic equivalents.

30 is the number of upright stones that originally encompassed the
Sarsen Circle, Stonehenge's best known feature.

Number Thirty (30): Organizing Principle of the Prime Number Sequence

The integer thirty (30), provides both vertical and horizontal structure
(a beautifully symmetrical superstructure, if you will) to the prime number sequence. For a detailed description and
graphic representation of this integral dimension of the primes, go to Prime Spiral Sieve.

References:

The work of exceptional amateur mathematician Philip G Jackson at www.simplicityinstinct.com
further corroborates the importance of the number 30. He has defined new prime number attributes and shows how what he calls "Prime Number Channels" provide a simple structure for understanding the pairing of primes in Goldbach's Strong Conjecture.

For an interesting discussion of the "magic" properties of number 30, check out Clif Droke's essay
here.

[Note: Logo graphic at header of this page incorporates image of "E8 graph of the Gosset 421 polytope as a
2-dimensional skew orthogonal projection inside Petrie polygon. It is an emulation of the hand drawn original by
Peter McMullen." The image is licensed by Creative Commons. For license terms
click here.]